Gelfand type problems involving the 1-Laplacian operator

Abstract

In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -\Delta_1u=\lambda f(u) &\hbox{in }\Omega\,;\\[2mm] u=0 &\hbox{on }\partial\Omega\,; \end{array} \right. \end{equation*} where $\Omega\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $\lambda \geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. It is proved the existence of a threshold $\lambda^*=\frac{h(\Omega)}{f(0)}$ (being $h(\Omega)$ the Cheeger constant of $\Omega$) such that there exists no solution when $\lambda>\lambda^*$ and the trivial function is always a solution when $\lambda\le\lambda^*$. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour of solutions to problems involving the $p$--Laplacian as $p$ tends to 1, which allows us to identify proper solutions through an extra condition.